Since the integrand is of the form where *p* > 1 and the interval of integration is [0,1], this integral diverges. Piece of cake, right? The hard part is remembering when you want *p* > 1 and when you want *p* < 1. There's a great trick for that: you can get by with remembering just one integral. The integral converges because gets close to the *x*-axis quickly as *x* approaches ∞. If you can remember this, then you can remember that because you have an example where *p* = 2 is greater than 1 and the integral converges. That means We switch between *p* > 1 and *p* < 1 when we change the interval of integration from [1,∞) to [0,1], so this means and That's the whole *p*-test, and all we had to remember was that the integral converges. We recommend taking your time when using the *p*-test. It can be easy to get mixed up, even if you know what you're doing! |